= Rotating accelerometer
A <rotating accelerometer> is an <accelerometer> whose sensing axes rotate with a moving component, such as a <gear> shaft. It can measure motion close to its source, but its output combines the point's actual acceleration with a changing projection direction. Treating its channels as fixed laboratory axes can therefore change the interpretation of a <gear vibration spectrum>. A NASA transmission experiment mounted a <microelectromechanical system> sensor package on a pinion and compared it with stationary housing measurements.<Evaluation of MEMS-Based Wireless Accelerometer Sensors in Detecting Gear Tooth Faults in Helicopter Transmissions by David G. Lewicki and others>[Printed pp.5–7, MEMS instrumentation and Figure7; p.19, channel combinations]{ref}
For a rigid shaft rotating in a plane, let its center have inertial acceleration $\mathbf a_O$. An attached point at radius $r$ has
$$
\mathbf a_P=\mathbf a_O+r\dot\omega\,\mathbf e_t-r\omega^2\mathbf e_r,
$$
where $\mathbf e_r,\mathbf e_t$ rotate with the shaft. The radial contribution is the <rotating-accelerometer centripetal term>. At constant $\omega=100\,\mathrm{rad/s}$ and $r=0.020\,\mathrm m$, its magnitude is $200\,\mathrm{m/s^2}$ even without vibration of the center. For the ideal <specific force> convention, the measured vector is $\mathbf a_P-\mathbf g$ projected onto each sensor axis, so <gravitational acceleration> can also produce a changing projection.
Suppose a laboratory acceleration $A\cos(2\pi f_m t)$ projects onto a sensor direction rotating at $f_r$. The product contains components at $f_m-f_r$ and $f_m+f_r$:
$$
\cos(2\pi f_mt)\cos(2\pi f_rt)
=\tfrac12\cos[2\pi(f_m-f_r)t]+\tfrac12\cos[2\pi(f_m+f_r)t].
$$
This measurement geometry can produce <modulation sidebands> without a damaged tooth. The NASA radial channel showed1854 and2060Hz around a1957Hz mesh frequency with103Hz rotation. Paired tangential channels also allowed different motion contributions to be separated by sums and differences.<Evaluation of MEMS-Based Wireless Accelerometer Sensors in Detecting Gear Tooth Faults in Helicopter Transmissions by David G. Lewicki and others>[Printed p.21, equations(1)–(2); Figure20, printed p.23]{ref} Correct interpretation therefore specifies mounting radius, orientation, <measurement calibration> and rotation timing, as well as sensor bandwidth.
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